Problems Open and Otherwise

Topological dynamics

Periodic points

  • Describe the set of periodic points or points with periodic itineraries in various settings (non-zero topological entropy, piecewise isometries,…)
  • Count them (find a nice zeta function)

Symbolic dynamics

  • describe its structure (e.g., by defining an abstract class to which the considered symbolic dynamics belong and which implies the considered properties)
  • relate it precisely to the PA dynamics (e.g., in which case does it have the same topological entropy?)

Entropy

  • dependence on the PA dynamics
  • computation from geometric and combinatorial data

Ergodic theory

SRB measures

  • Simplify Tsujii's general construction
  • Find something like it for the piecewise hyperbolic situation, maybe using Baladi-Gou√ęzel recent work (prove first a generic result?)
  • Prove Palis conjecture on SRB measures for PA dynamics, for instance for the Lozi family

Maximal entropy measures

  • Find in which settings:
    • they always exist (e.g., homeomorphisms in higher dimension?)
    • they form a finite-dimensional simplex
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